The optimized orbit transfer of a space vehicle, revolving initially around the primary, in a similar orbit to that of the Earth around the Sun, in an elliptic trajectory, to another similar elliptic orbit of an adequate outer planet is studied in this paper. We assume the elements of the initial orbit to be that of the Earth, and the elements of the final orbit to be that of an outer adequate planet, Mars for instance. We consider the case of two impulse generalized Hohmann non coplanar orbits. We need noncoplanar (plane change) maneuvers mainly because: 1) a launch-site location restricts the initial orbit inclination for the vehicle; 2) the direction of the launch can influence the amount of velocity the booster must supply, so certain orientations may be more desirable; and 3) timing constraints may dictate a launch window that isn’t the best, from which we must make changes[3]. We used the Lagrange multipliers method to get the optimum of the total minimum energy required ΔVT , by optimizing the two plane change angles 1 and 2, where 1 is the plane change at the first instantaneous impulse at peri-apse, and 2 the plane change at the second instantaneous thrust at apo-apse. We adopt the case of Earth - Mars, as a numerical example.
We briefly discuss the notion of the Lagrange multiplier for a linear constraint in the Hilbert space setting, and we prove that the pressure p appearing in the stationary Stokes equations is the Lagrange multiplier of the constraint div u = 0.
The article presents the use of Lagrange multipliers functions to inventory stocks management under the limited capital conditions. The example verified the method based on the process analysis, the nonlinear programming and the Solver application.
PL
W artykule przedstawiono możliwość wykorzystania mnożników Lagrangea do zarządzania zapasami w warunkach ograniczonego kapitału. Zaproponowano weryfikację ww. metody w oparciu o programowanie nieliniowe i aplikację Solver.
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Laminated glass structures are formed by stiff layers of glass connected with a compliant plastic interlayer. Due to their slenderness and heterogeneity, they exhibit a complex mechanical response that is difficult to capture by single-layer models even in the elastic range. The purpose of this paper is to introduce an efficient and reliable finite element approach to the simulation of the immediate response of laminated glass beams. It proceeds from a refined plate theory due to Mau (1973), as we treat each layer independently and enforce the compatibility by the Lagrange multipliers. At the layer level, we adopt the finite-strain shear deformable formulation of Reissner (1972) and the numerical framework by Ibrahimbegović and Frey (1993). The resulting system is solved by the Newton method with consistent linearization. By comparing the model predictions against available experimental data, analytical methods and two-dimensional finite element simulations, we demonstrate that the proposed formulation is reliable and provides accuracy comparable to the detailed two-dimensional finite element analyzes. As such, it offers a convenient basis to incorporate more refined constitutive description of the interlayer.
In this paper, we have studied a scheduling problem associated with air-traffic control, in which a set of airerafts are about to land on a single runway. The objectives are to maximize the minimum time elapsed between any two consecutive landings, minimize the maximum number of times a plane enters a holding time, likewise including the priority of aircraft landings. We use the mixed integer three dimensional problem formulation that allows us to solve large instances of the general problem. Our numerical results validated the effectiveness of the proposed method.
PL
W artykule przedstawiono problem szeregowania samolotów podchodzących do lądowania i startujących na drodze startowej. Przyjęto, że celem szeregowania jest minimalizacja maksymalnej liczby samolotów oczekujących w powietrzu na lądowanie przy uwzględnieniu ich priorytetów. Problem sformułowano jako trójwymiarowe mieszane programowanie całkowitoliczbowe. Dostarczone przykłady potwierdziły efektywność proponowanej metody rozwiązania tego problemu.
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In this paper, we propose to introduce the single and double Lagrange multipliers approaches in the case of the finite element method (FEM). These approaches allow non-conforming meshes to be linked together. The methods introduced are developed in the case of a magnetostatic problem solved by the scalar potential formulation. An application is studied and the results obtained by both approaches are compared.
PL
W artykule przedstawiono zastosowanie pojedynczych i podwójnych mnożników Lagrangea stosowanych w metodzie elementów skończonych. To podejście pozwala na połączenie niezgodnych siatek. Metodę rozwinięto dla problemu magnetostatycznego rozwiązywanego z użyciem potencjału skalarnego. Porównano wyniki otrzymane z zastosowaniem proponowanej metody i metod klasycznych.
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Artykuł dotyczy zagadnienia swobodnych drgań poprzecznych belek o skokowo-zmiennym polu przekroju poprzecznego w połączeniu z dodatkowymi elementami dyskretnymi typu: sprężyna transakcyjna i rotacyjna, oscylator harmoniczny, masa skupiona oraz dodatkowe zamocowania. Do sformułowania i rozwiązania zagadnienia wykorzystano formalizm mnożników Lagrange'a.
EN
The modeling and solution of the free lateral vibration problem of stepped beam with additional discrete like: translational and rotational spring, harmonic oscillator, mass and fix elements, has been presented. The procedure has been based on the Lagrange multiplier formalism. Additionally, the presented paper contains the numerical example illustrating the use of the derived expressions.
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